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Author(s)

First Name Middle Name Surname Role Email

Majdi 1 R. Abou Najm ASABE Member

[email protected]

Chadi2 S. El mohtar [email protected]

Rabi 1 H. Mohtar ASABE Member

[email protected]

Vincent 2 P. Drnevich [email protected]

Erik 3 Braudeau [email protected]

Affiliation

Organization Address Country

1 Purdue University

Purdue University Department of Agricultural & Biological Engineering 225 South University Street West Lafayette, IN 47907-2093

USA

2 Purdue University

School of Civil Engineering 550 Stadium Mall Drive West Lafayette, IN 47907-2051

USA

3 IRD SeqBio-IRD Ensam 2 place Viala, Bâtiment 12 34060 Montpellier cedex 2

France

Publication Information

Pub ID Pub Date

084952 2008 ASABE Annual Meeting Paper


An ASABE Meeting Presentation Paper Number: 084952

Improving TDR Measurements through Accounting for Soil Shrinkage Properties

Majdi R. Abou Najm

Agricultural & Biological Engineering, Purdue University, [email protected]

Chadi El mohtar

Civil Engineering, Purdue University, [email protected]

Rabi H. Mohtar

Agricultural & Biological Engineering, Purdue University, [email protected]

Vincent P. Drnevich

Civil Engineering, Purdue University, [email protected]

Erik Braudeau

Institut de Recherche pour le Développement (IRD), France

Written for presentation at the 2008 ASABE Annual International Meeting

Sponsored by ASABE Rhode Island Convention Center

Providence, Rhode Island June 29 – July 2, 2008

Mention any other presentations of this paper here, or delete this line.

Abstract. Time domain reflectometry (TDR) is a widely used tool for indirect measurement of soil moisture content. Empirical formulation is used to link the apparent dielectric constant of soil to the volumetric moisture content (Topp’s Equation: Topp et al., 1980) or gravimetric moisture content as a function of soil bulk density (Siddiqui and Drnevich Equation: Siddiqui and Drnevich, 1995). This paper introduces a methodology to account for soil volume change by integrating the true bulk


density of the soil into the measurements using the soil Shrinkage Curve (specific volume (cm3/g) versus gravimetric water content). Thus, bulk density becomes a non-constant parameter that can be calculated as a function of the soil water content in the Siddiqui and Drnevich Equation. Experimental evidence demonstrates accounting for soil shrinkage improves the accuracy of TDR measured moisture contents and allows for estimating the shrinkage curve. Direct water content calculation for the Chalmers soil was compared to water contents from TDR readings with and without shrinkage corrections; those with shrinkage corrections showed significantly improved accuracy in TDR-determined soil moisture.

Keywords. TDR, soil structure, shrinkage properties, shrinkage curve, drying process, water content.

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Introduction

Achieving accurate measurements of the soil moisture content is an active area of research and experimentation (Noborio, 2001, Evett and Parkin, 2005, Robinson et al., 2003 and 2008). Various direct and indirect methods for the measurement and estimation of the water content have been developed including oven-drying, as well as the in-situ electrical resistance, neutron probe, tensiometers, and time domain reflectometry (TDR). While the oven-drying method represents the only direct method for the measurement of soil water content, non-destructive in-situ measurements require empirical or constitutive correlations between water content and soil response to certain sensor stimuli like change in concentration of slowed neutrons, electrical conductivity, or the soil apparent dielectric constant, Ka.

TDR, a widely accepted and increasingly accurate measurement method of soil water content, relies on the accurate measurement of Ka given its sensitivity to the amount of water in the soil (Topp et al., 1980, Siddiqui et al., 2000, Drnevich et al., 2005, Yu and Drnevich, 2004). This sensitivity comes from the large difference between Ka for water (around 81), soil particles (between 2 and 7) and air (around 1) (Drnevich et al., 2005). Most TDR equipment uses the Topp equation (Equation 1: Topp et al., 1980) in the estimation of the volumetric water content using a cubic equation in terms of Ka. Siddiqui and Drnevich (1995) derived a relationship between Ka, dry bulk density (ρd), and the gravimetric water content of the soil, w (Equation 2):

kkk aaaTDR362422

103.4105.51092.2103.5

Equation 1

w

a

d

a bwK

Equation 2

where ρw is the density of water. a and b are soil-dependent parameters with a related to ρd and the Ka of the dry soil and b related to the increase Ka with increase in the pore fluid.

While the effect of ρd on the Siddiqui and Drnevich Equation has been studied in the context of compaction energy due to its direct relevance to geotechnical applications (Drnevich et al., 2005), the effect of change in ρd due the soil shrinkage has not been addressed yet. Accordingly, the effect of structural interaction and internal stress (or energy) change due to shrinkage on Ka and TDR response is not fully understood. Moreover, various studies on clayey and silty soils have observed considerable effect for temperature, clay content, salinity, saturating cations, frequency, pH, and CEC on the water content predictability using Ka through empirical formulation (Saarenketo, 1998, Ponizovsky et al., 1999, Ishida and Makino, 1999, Logsdon and Laird, 2004, Kelleners et al., 2004, Chen and Or, 2006, Robinson et al., 2008). Such observations call for more physically based understanding of the relationship between Ka and water content.

This paper assesses the performance of the Topp’s as well as the Siddiqui and Drnevich Equations in predicting the volumetric and gravimetric water contents of a drying silty clay loam soil. The paper then introduces a methodology to account for the soil hydro-structural change

effect by introducing the real soil specific volume, 1/d, as a function of the gravimetric water content, w, (obtained from the soil Shrinkage Curve) into Equation 2 that is used for the measurement of w. Thus, bulk density becomes a non-constant parameter that can be calculated as a function of the soil water content using the experimental procedure defined in Braudeau et al., 1999. Finally, the paper presents a new methodology to reconstruct/estimate the shrinkage curve using only five data points to calibrate the TDR estimates of gravimetric and volumetric water contents. The new methodology is tested by calibrating the TDR with data from

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a soil sample and then using the calibration parameters to reconstruct the shrinkage curve (of another experiment) with a very close fit.

Materials and Methods

To assess the effect of soil volume change due to shrinkage, continuous measurements of volumetric water content as estimated by the TDR (TDR ) must be compared with simultaneous

and direct measurement of the real gravimetric water content wReal, as well as simultaneous calculation of 1/ρd, taking into account the relationship between gravimetric (w) and volumetric

( ) water contents (Equation 3). This becomes an ideal setup for a typical shrinkage curve

experiment (Braudeau et al., 1999, Braudeau et al., 2004) with the addition of the TDR probe and equipment.

d

ww Equation 3

Accordingly, comparisons can be made between the measured (wReal) and predicted gravimetric water content (wTDR), derived from TDR using Equation 3 and results of ρd from the shrinkage

curve. Similarly, comparisons can be made between the measured volumetric water content

(Real ), (derived from wReal using Equation 3 and results of ρd from the shrinkage curve) and

predicted TDR . The effect of change in ρd due to shrinkage on the TDR methodology for

measuring the soil water content, can be assessed by comparing the performance of TDR equipment using: (1) built-in parameters vs. soil-specific parameters, (2) a single value of ρd and (3) the specific volume, 1/ρd, as function of wReal (according to the Shrinkage Curve).

Procedure

The procedure followed in the experiments (Figure 1) was:

1. Soil preparation: Disturbed soil was sieved using sieve #4 and mixed with water to the

desired water content (near saturation). The soil mixture was sealed and kept for a week in humid room to allow the water content to equilibrate.

2. Specimen base: An acrylic sheet, randomly perforated (with 3/16-inch (4.7-mm) diameter holes) was used as the base for the soil specimen. The sheet was overlaid with randomly-cut small geotextile sheets (Figure 1) to allow for water exfiltration or evaporation from the bottom of the specimen while preventing the loss of soil as water escapes the soil.

3. Sample preparation: The soil specimen was prepared by uniformly compacting soil in multiple layers (6 to 8) inside a PVC tube (15cm long and 7.73cm in diameter), used only to confine the soil. A thin layer of oil was placed on the inside of the PVC tube to reduce the friction between the PVC and the soil.

4. TDR Equipment: A TRASE TDR unit (SoilMoisture Equipment Corp) was used in these experiments. The TDR three-rod probe provided with the TRASE TDR was inserted into the soil specimen and the TDR equipment built-in software was used to collect Ka and

TDR readings every 10 minutes for the test duration which lasted approximately four

weeks. 5. Calculation of wReal: The soil specimen was placed on an electronic balance connected

to a data acquisition system (using WinWedge software) and the data acquisition system

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was programmed to measure the mass of the soil specimen at 10 minute intervals. At the end of the experiment, the gravimetric water content of the specimen was determined by oven drying. Using the last gravimetric water content as a starting point, the water contents of the soil specimen was back calculated as a function of time by use of the measured specimen mass.

6. Calculation of ρd: The change in specimen length due to shrinkage was measure by using an LVDT. Assuming isotropic shrinkage of the specimen (Braudeau et al., 1999, Braudeau et al., 2004), and using the soil mass obtained from the previous step, the bulk density (or specific volume) can be calculated as function of time. To avoid any punching effect by the LVDT core, the core was placed on a geotextile sheet instead of directly being placed on the soil.

Figure 1. Schematic of the experiment procedure.

It was a challenge to minimize the water content gradients in the soil specimen because those gradients could affect the TDR readings. This objective was partially achieved by allowing the soil specimen to dry at room temperature from the top, bottom, and side boundaries (as the soil shrinks, the soil loses contact with the PVC tube). The design of the soil specimen base and the use of oil at the PVC tube allowed for drying from the bottom and boundaries, respectively.

In nutshell, the shrinkage curve experimental setup is used as a control for the purpose of this study providing wReal and ρd as function of time. The TDR equipment on the other hand provides

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TDR (volumetric water content estimated by the TDR) and Ka. The objective of this study is

assess the accuracy of TDR (and the wTDR derived there from) as provided by the TDR

equipment to improve those values over those by solving for soil-specific constants (Equations 1 and 2) using a single value of ρd (wTDR_Single ρd), through solving for soil-specific constants using instantaneous values of ρd as predicted and measured by the shrinkage curve (wTDR_ShC).

Materials

Samples of the Chalmers clay loam soil were collected from the Agronomy Center for Research and Education (ACRE) at Purdue University. The Chalmers soil consists of 15% sand, 45% silt, and 40% clay. A complete list of soil properties is provided by Abou Najm et al. (2008). The use of disturbed samples (as explained in the previous section) instead of undisturbed field-collected specimens was favored since laboratory-made soil specimens can be prepared with more uniformity, thus avoiding the presence of large stones, non-soil residues, and major cracks and discontinuities.

Experimental Results

Three experiments were conducted for the same soil following the procedure explained in the previous section. The average duration of each experiment was around 27 days. At the end of each experiment, data was downloaded and analyzed to obtain: TDR and Ka (from the TDR

unit), wReal (from electronic balance), and ρd (from electronic balance and LVDT). wTDR and alRe

were then derived using TDR , wReal, ρd and Equation 3. Figure 2 shows the shrinkage curve of

the Chalmers soil from the first experiment (experiment 1). The shrinkage curve is presented in specific volume, 1/ρd (cm3/g), as function of w.

Figure 2. The shrinkage curve of the Chalmers Soil from experiment 1.

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Comparison with Topp’s Equation

Figure 3 shows a comparison between the non-calibrated TDR readings (TDR and wTDR: using

Topp’s Equation) and the real gravimetric (wReal) and volumetric ( alRe ) water contents for

experiment 1 as obtained by the shrinkage curve. It is clear from Figure 3 that the TDR predictions (TDR and wTDR) were not very representative of the real water content of the soil

specimen ( alRe and wReal). The main reasons behind the differences in the water contents are:

(1) the relatively high clay content of the soil causing dielectric dispersion or change in permittivity as function of frequency (Saarenketo, 1998, Ishida and Makino, 1999, Chen and Or, 2006, Robinson et al., 2008), (2) the high shrinkage range of the soil (ρd increased from around 1.30 g/cm3 to 2.0 g/cm3as the soil dried from wReal=40.01% to wReal=6.75%), and (3) the empirical nature of the Topp’s equation (Topp et al., 1980).

Figure 3. Comparison between the non-calibrated TDR readings and the real gravimetric (kg/kg) and volumetric (m3/m3) water content values as estimated by the shrinkage curve

experiment.

The objective of this study is to improve the predictability of TDR equipment for the water content of swelling soils. Thus, the coefficients of the Topp’s equation (Equation 1) were calibrated through optimization that minimizes the sum of the square of the error between the real and predicted volumetric water content (Equation 4.a). Similar empirical form of the Topp equation was obtained for the gravimetric water content as function of Ka by minimizes the sum of the square of the error between the real and predicted gravimetric water content (Equation 4.b). Figure 4 shows a comparison between the calibrated TDR readings and the real

wReal: real gravimetric water content obtained by back

calculation of weighing balance readings;

TDR : volumetric water content as obtained from TDR.

Topp : Application of Equation 1 to obtain volumetric

water content using Ka values;

alRe : Obtained by multiplying WReal by the bulk density

(Equation 3 assuming ρw = 1); wTDR,Topp: gravimetric water content obtained from the

Topp’s equation and bulk density (Theta Topp’s/bulk density).

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gravimetric and volumetric water content values showing an excellent fit between the real and the four-parameter Topp’s equation (after calibration).

kkk aaacalibratedTDR352342

, 1056.41084.11001.51079.9 Equation 4.a

kkkw aaacalibratedTDR352322

, 103.3107.11018.11055.9 Equation 4.b

Figure 4. Comparison between the calibrated TDR readings and the real gravimetric (kg/kg) and volumetric (m3/m3) water content values as estimated by the shrinkage curve experiment.

Comparison with Siddiqui and Drnevich Equation

In addition to comparison with the Topp’s Equation, the Siddiqui and Drnevich Equation was utilized in two ways: (1) using a single value of ρd obtained from the shrinkage curve and (2) using the real ρd as function of w.

First, the a and b parameters of Equation 2 were obtained by minimizing the square of the error (wReal - wTDR_Single BD)2 using the Microsoft Excel optimization solver assuming constant value of bulk density. An interesting observation was that the optimized values of wTDR_Single BD followed the same curve irrespective of the choice of ρd (or BD). However, the a and b parameters decreased as ρd increased (or wReal decreased). Table 1 shows the results of the a and b parameters for three ρd values taken at wReal = 40%, 20% and 6.75%. Figure 5 shows a

0

10

20

30

40

50

60

0 5 10 15 20 25 30

Time (Days)

Gra

vim

etr

ic W

ate

r C

onte

nt

(g/g

)

theta Real Theta Topp Calibrated WReal W Topp Calibrated

alRe : Obtained by multiplying WReal by the bulk density

(Equation 3 assuming ρw = 1); Theta Topp Calibrated: Application of optimization to

derive soil specific parameter of Equation 1 to obtain volumetric water content using Ka values; WReal: real gravimetric water content obtained by back

calculation of weighing balance readings; W Topp Calibrated: gravimetric water content obtained from Theta Topp Calibrated and ρd (Theta Topp’s/ ρd).

8

considerable improvement in the predictability of the water content using Equation 2 (for experiment 1). However, error (wReal- wTDR_Single BD)2 was still within five water content percentage points over a considerable portion of the experimental range.

Table1. Variation in the a and b parameters for experiment 1. Assuming

constant ρd at WReal = 40%

Assuming constant ρd at

WReal = 20%

Assuming constant ρd at WReal = 6.75%

Assuming dynamic ρd as predicted by

the ShC

a 1.45 1.12 1.03 0.60

b 5.79 4.45 4.10 7.56

Further improvement of TDR prediction using the Siddiqui and Drnevich Equation was achieved by solving for a and b after substituting the corresponding value of ρd (function of w) to obtain gravimetric water content predictions using Ka, ρd, and calibrated a and b values (wTDR_ShC). The a and b were also obtained by minimizing the square of the error (wReal- wTDR_ShC)2 (Table 1). Figure 5 shows an excellent match between wReal and wTDR_ShC. This shows the importance of the shrinkage properties of soils in predicting the water content using the two-parameter Siddiqui and Drnevich Equation 2. Statistically, the coefficient of determination, R2, improved from 0.9664 to 0.9846 when changing from single bulk density value to a variable value that is a function of w. Similarly, the mean absolute error improved from 1.50 to 0.85 percentage points for gravimetric water contents when changing from single bulk density value to values as function of w.

To demonstrate the uniqueness of this approach for a given soil type, the analysis performed from the first experiment (experiment 1) was tested on experiments 2 and 3 using the parameters from experiment 1. Here, uniqueness is defined as the repetitiveness of the obtained parameters from one experiment to the others if the same soil type is used. Results will be discussed in the next section.

Discussion

Results of the second and third experiments were analyzed similar to experiment 1 (the three experiments utilized three replicates of the same soil). Results were first analyzed with the complete dataset for each experiment. The shrinkage process changes quickly at high water contents and becomes very slow at low water contents (Figure 2). This means that all the datasets in the three experiments have more data points per unit of wReal at low water contents and this may affect the results of the optimization process, whether calibrating for the soil-specific four parameters of the Topp Equation or for a and b for the Siddiqui and Drnevich Equation. Thus, to provide a uniform distribution of the dataset across the range of water contents, datasets were rearranged by considering only one representative data point for each half percentage point of wReal.

The uniqueness of this method was tested by substituting the values of a and b obtained for experiment 1 to experiments 2 and 3. Figure 6 shows a comparison between wReal and wTDR_ShC. An excellent match between wReal and the predicted wTDR_ShC is observed for the three experiments based on using the a and b values of experiment 1. However, it was observed that at wReal values higher than 39%, the wTDR_ShC consistently over-predicted wReal. This means that above a certain threshold value of wReal, Equation 2, or the procedure adopted in this analysis, fails to predict the gravimetric water content of the soil. Thus a physical meaning for the

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threshold wReal value needs to be established or else the predictability of water content by Equation 2 may not be unique or reliable.

Figure 5. The performance of the Siddiqui and Drnevich Equation in estimating W demonstrated through a comparison between WReal and WTDR_SingleBD (as estimated by Equation 2 and a single bulk density value) and WTDR_ShC (as estimated by Equation 2 and a dynamic bulk

density value) for experiment 1.

Figure 6. Results of the three experiments showing the effectiveness of accounting for the change in the soil bulk density in improving TDR predictions. The a and b of experiment 1 were

used for experiment 2 and experiment 3.

wTDR_ShC: Calibrated Siddiqui and Drnevich TDR reading estimate using dynamic bulk density value as predicted by the shrinkage curve. wTDR_Single BD: calibrated Siddiqui and Drnevich TDR reading estimate using single bulk density value in Equation 1.

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The search for a physical explanation to the threshold value above which Equation 2 over predicts wReal triggered the attention towards the shrinkage curve. By mapping the shrinkage curve over wReal vs. wTDR_ShC, it was observed (Figure 7) that the same threshold value above which wTDR_ShC over predicts wReal is the air entry point of the macro structure in the shrinkage curve (as explained by Braudeau et al., 2004). This point is defined when the slope of the shrinkage curve becomes less than one thus implying that the soil is no more at saturation. Physically, it means that the change in volume of the soil is less than the change in wReal with some of the pores filled with water becoming filled with air (instead of being totally lost to volume change). For TDR, this means that Equation 2 may best predict the gravimetric water content of the soil when the soil is at or below the saturation limit. At much higher water contents, the soil may act as a slurry more than a soil and it is assumed that this is the reason why TDR over predicted the water content in that range.

Relevance to the Industry

Relevance to industry comes through incorporating the effect of the soil structural interactions by accounting for the soil dynamic shrinkage/swelling behavior in the calculation of the soil bulk

Range of W above which TRD

over predicts WReal. This point

happens to be the air entry

point in the Shrinkage curve,

i.e. the point above which

shrinkage curve has a slope

equal to 1.

Range of W below

which TRD can

highly predict WReal

using Equation 1.

Figure 7. The use of the Shrinkage Curve to define threshold for TDR predictability.

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density. This can be achieved by allowing users to enter the unique shrinkage/swelling curve parameters as part of the input in the TDR equipment. This however requires one further step that is to translate the x-axis of the shrinkage curve from wReal to Ka. The reason is that TDR measures Ka directly and infers TDR or wTDR by relating Ka to the volumetric or gravimetric

water content through Equations 1, 2, and/or 3. Thus, the TDR equipment would require all

models to be a function of Ka rather than w or . Moreover, accounting for volume change

improved the predictability of the gravimetric water content by the TDR equipment simply because Equation 2 contained the bulk density term (which makes it more physically based than Topp’s Equation which is totally empirical). Accordingly, incorporating volume change properties of the soil in TDR equipment requires mapping the soil shrinkage/swelling properties into a unique relationship with Ka as demonstrated in Figure 8.

It is proposed that this research be extended to the swelling direction. This can be achieved by slightly modifying the experimental procedure by allowing for adding water to a dry soil specimen, and again taking TDR and electronic balance readings at fixed time intervals. In all cases, shrinkage/swelling behavior can be incorporated into the new TDR equipment by relating those properties to Ka for a given soil and include them in the calculations for the next generation of TDR equipment.

Figure 8. Shrinkage Curve as function of Ka instead of WReal.

Predicting the Shrinkage Curve

Gravimetric and volumetric water contents interact in a unique relationship (Equation 3) that can be re-written as:

lumeSpecificVow

dw

1 Equation 5

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Thus, if w and can be predicted by the TDR, then the specific volume of the soil can be

calculated using Equation 5.

To test this hypothesis, five points from experiment 3 were considered (Table 2). The values for

w and vs. Ka were used to construct a cubic polynomial constitutive relationship similar in

form to the Topp’s Equation (Figure 9). This allows for the prediction of w and for every value

of Ka as explained in Equations 6 and 7:

kkk aaaCalibratedTDR352332

, 100.7107.2103.4106.9 Equation 6

kkkW aaaCalibratedTDR352342

, 100.2101.11000.21015.4 Equation 7

Using Equation 5, estimates of the specific volume of the soil in experiment 1 were plotted against the gravimetric water content (i.e. the shrinkage curve) and compared to the real shrinkage curve of experiment 1 (Figure 10). Although results are not a perfect match, the fact that such an estimate of shrinkage curve of experiment 1 was obtained from calibration of five data points of a different soil replicate (experiment 3) make this method a promising tool to estimate the bulk density of the soil, and definitely an application worth considering for TDR equipment.

Figure 9. Calibration of the gravimetric and volumetric water content predictions using Topp’s-like polynomial equations estimated from five experimental points in experiment 3.

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Figure 10. Comparison between the real shrinkage curve of experiment 1 and the predicted curve based on calibrated gravimetric and volumetric water content predictions using Topp’s-

like polynomial equations estimated from only five experimental points in experiment 3.

Table2. Five data points from experiment 3 used to estimate the shrinkage curve of experiment 1.

Point Ka WReal alRe

1 24.7 37.5 55.3

2 23.7 35.0 53.7

3 14.1 20.0 37.6

4 8.6 10.5 20.8

5 5.2 7.0 13.9

Conclusion

TDR equipment provides accurate predictions of the soil volumetric and gravimetric water content through direct measurement of the apparent dielectric constant of the soil, Ka. Experimental evidence was provided to demonstrate the effect of soil shrinkage on improving the accuracy of water contents from TDR measurements. Direct water content calculation for the Chalmers soil was compared to TDR readings with and without shrinkage correction and showed considerable improvements in soil moisture by accounting for soil shrinkage. This work provides experimental evidence on the importance of incorporating the volume change behavior

of soils into TDR equipment for improving the predictability of volumentric water content ( ) or

gravimetric water content (w) as follows:

14

1. Comparison between direct measurements of or w and indirect measurements using

TDR’s factory built-in parameters showed unsatisfactory accuracy for swelling soils.

2. Improving the predictability by solving for soil-specific parameters assuming a single bulk density value (thus no volume change) showed considerable improvement, but errors up to five water content percentage points at high and low extremes in water contents occurred with the Siddiqui and Drnevich Equation.

3. Accounting for volume change by substituting the corresponding value of ρb in the Siddiqui and Drnevich Equation for each w estimation demonstrated very accurate predictability of gravimetric water content by the TDR equipment below a certain threshold.

4. This threshold was defined as the point of soil saturation in the shrinkage curve, above which the soil behaves as slurry.

5. By defining soil-specific parameters for the estimation of volumetric and gravimetric water contents, the bulk density (or specific volume) of the soil can be estimated with considerable accuracy (Equation 5).

Finally, it was proposed that the outcome of this research will form the first steps towards next generation of TDR data analysis algorithms that can account for volume change properties of the soil by allowing for user-input of the shrinkage/swelling properties and parameters of the soil.

Acknowledgements

This research was supported by a USDA grant No. 2005-03338 and a research grant from the French Embassy office of Science and Technology in Chicago. The authors would also like to acknowledge Janet Lovell, Director of the Geotechnical Lab in the School of Civil Engineering for providing the resources to perform the tests.

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Paper No: 200000 - University of Texas at AustinAn ASABE Meeting Presentation Paper Number: 084952 Improving TDR Measurements through Accounting for ... (IRD), France Written for presentation